# HG changeset patch # User Jens Rasch # Date 1245512650 -3600 # Node ID 8ae379ab614c7120d7745d16440f24f98c82d1e0 # Parent a3427ba1dfd0b77e1b600cc0ab8180b1408d46e4 Removed all Latex formulas and replaced them with text formulas to comply with the ReST documentation system. diff -r a3427ba1dfd0 -r 8ae379ab614c sage/functions/wigner.py --- a/sage/functions/wigner.py Sun Jun 14 12:08:42 2009 +0100 +++ b/sage/functions/wigner.py Sat Jun 20 16:44:10 2009 +0100 @@ -75,9 +75,7 @@ def Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3,prec=None): r""" - Calculate the Wigner 3j symbol - - \left({j_1 \atop m_1} {j_2 \atop m_2} {j_3 \atop m_3} \right) + Calculate the Wigner 3j symbol Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3) NOTES: @@ -86,33 +84,23 @@ - invariant under any permutation of the columns (with the exception of a sign change where $J:=j_1+j_2+j_3$): - \begin{eqnarray} - \left({j_1 \atop m_1} {j_2 \atop m_2} {j_3 \atop m_3}\right) - &=& - \left({j_3 \atop m_3} {j_1 \atop m_1} {j_2 \atop m_2}\right) - =\left({j_2 \atop m_2} {j_3 \atop m_3} {j_1 \atop m_1}\right) - \qquad \mbox{cyclic} - &=& - (-)^{J}\left({j_3\atop m_3} {j_2\atop m_2} {j_1\atop m_1}\right) - =(-)^{J}\left({j_1\atop m_1} {j_3\atop m_3} {j_2\atop m_2}\right) - =(-)^{J}\left({j_2\atop m_2} {j_1\atop m_1} {j_3\atop m_3}\right) - \qquad\mbox{anti-cyclic} - \end{eqnarray} + Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3) + =Wigner3j(j_3,j_1,j_2,m_3,m_1,m_2) + =Wigner3j(j_2,j_3,j_1,m_2,m_3,m_1) + =(-)^J Wigner3j(j_3,j_2,j_1,m_3,m_2,m_1) + =(-)^J Wigner3j(j_1,j_3,j_2,m_1,m_3,m_2) + =(-)^J Wigner3j(j_2,j_1,j_3,m_2,m_1,m_3) - invariant under space inflection, i. e. - \begin{eqnarray} - \left({{j_1}\atop{m_1}} {{j_2}\atop{m_2}} {{j_3}\atop{m_3}}\right) - = - (-)^{J} - \left({j_1 \atop -m_1} {j_2 \atop -m_2}{j_3 \atop -m_3}\right) - \end{eqnarray} + Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3) + =(-)^J Wigner3j(j_1,j_2,j_3,-m_1,-m_2,-m_3) - symmetric with respect to the 72 additional symmetries based on the work by [Regge58] - zero for $j_1$, $j_2$, $j_3$ not fulfilling triangle relation - - zero for ${m_1}+{m_2}+{m_3}!= 0$ + - zero for $m_1+m_2+m_3!= 0$ - zero for violating any one of the conditions $j_1\ge|m_1|$, $j_2\ge|m_2|$, $j_3\ge|m_3|$ @@ -258,19 +246,15 @@ r""" Calculates the Clebsch-Gordan coefficient - $\left< j_1 m_1 \; j_2 m_2 | j_3 m_3 \right>$ + $< j_1 m_1 \; j_2 m_2 | j_3 m_3 >$ NOTES: The Clebsch-Gordan coefficient will be evaluated via its relation to Wigner 3j symbols: - - \begin{eqnarray} - \left< j_1 m_1 \; j_2 m_2 | j_3 m_3 \right>= - (-1)^(j_1-j_2+m_3) \sqrt(2j_3+1) - \left({j_1 \atop m_1} {j_2 \atop m_2} {j_3 \atop -m_3}\right) - \end{eqnarray} + < j_1 m_1 \; j_2 m_2 | j_3 m_3 > + =(-1)^(j_1-j_2+m_3) \sqrt(2j_3+1) Wigner3j(j_1,j_2,j_3,m_1,m_2,-m_3) See also the documentation on Wigner 3j symbols which exhibit much higher symmetry relations that the Clebsch-Gordan coefficient. @@ -402,11 +386,8 @@ NOTES: The Racah symbol is related to the Wigner 6j symbol: - \begin{eqnarray} - \left({j_1 \atop j_4} {j_2 \atop j_5} {j_3 \atop j_6} \right) - = - (-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6) - \end{eqnarray} + Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6) + =(-1)^(j_1+j_2+j_4+j_5) W(j_1,j_2,j_5,j_4,j_3,j_6) Please see the 6j symbol for its much richer symmetries and for additional properties. @@ -491,49 +472,33 @@ def Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6,prec=None): r""" - Calculate the Wigner 6j symbol - - \left({j_1 \atop j_4} {j_2 \atop j_5} {j_3 \atop j_6} \right) + Calculate the Wigner 6j symbol Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6) NOTES: The Wigner 6j symbol is related to the Racah symbol but exhibits more symmetries as detailed below. - \begin{eqnarray} - \left({j_1 \atop j_4} {j_2 \atop j_5} {j_3 \atop j_6} \right) - = - (-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6) - \end{eqnarray} + Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6) + =(-1)^(j_1+j_2+j_4+j_5) W(j_1,j_2,j_5,j_4,j_3,j_6) The Wigner 6j symbol obeys the following symmetry rules: - Wigner $6j$ symbols are left invariant under any permutation of the columns: - \begin{eqnarray} - \left({j_1 \atop j_4} {j_2 \atop j_5} {j_3 \atop j_6} \right) - &=& - \left({j_3 \atop j_6} {j_1 \atop j_4} {j_2 \atop j_5} \right) - =\left({j_2 \atop j_5} {j_3 \atop j_6} {j_2 \atop j_4} \right) - \qquad \mbox{cyclic} \\ - &=& - \left({j_3 \atop j_6} {j_2 \atop j_5} {j_1 \atop j_4} \right) - =\left({j_1 \atop j_4} {j_3 \atop j_6} {j_2 \atop j_5} \right) - =\left({j_2 \atop j_5} {j_1 \atop j_4} {j_3 \atop j_6} \right) - \qquad \mbox{anti-cyclic} - \end{eqnarray} + Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6) + =Wigner6j(j_3,j_1,j_2,j_6,j_4,j_5) + =Wigner6j(j_2,j_3,j_1,j_5,j_6,j_4) + =Wigner6j(j_3,j_2,j_1,j_6,j_5,j_4) + =Wigner6j(j_1,j_3,j_2,j_4,j_6,j_5) + =Wigner6j(j_2,j_1,j_3,j_5,j_4,j_6) - They are invariant under the exchange of the upper and lower arguments in each of any two columns, i. e. - \begin{eqnarray} - \left({j_1 \atop j_4} {j_2 \atop j_5} {j_3 \atop j_6} \right) - = - \left({j_1 \atop j_4} {j_5 \atop j_2} {j_6 \atop j_3} \right) - = - \left({j_4 \atop j_1} {j_2 \atop j_5} {j_6 \atop j_3} \right) - = - \left({j_4 \atop j_1} {j_5 \atop j_2} {j_3 \atop j_6} \right) - \end{eqnarray} + Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6) + =Wigner6j(j_1,j_5,j_6,j_4,j_2,j_3) + =Wigner6j(j_4,j_2,j_6,j_1,j_5,j_3) + =Wigner6j(j_4,j_5,j_3,j_1,j_2,j_6) - additional 6 symmetries [Regge59] giving rise to 144 symmetries in total @@ -631,14 +596,7 @@ def Wigner9j(j_1,j_2,j_3,j_4,j_5,j_6,j_7,j_8,j_9,prec=None): r""" Calculate the Wigner 9j symbol - - \left\{ - \begin{matrix} - {j_1} & {j_2} & {j_3} \\ - {j_4} & {j_5} & {j_6} \\ - {j_7} & {j_8} & {j_9} - \end{matrix} - \right\} + Wigner9j(j_1,j_2,j_3,j_4,j_5,j_6,j_7,j_8,j_9) ALGORITHM: @@ -756,16 +714,11 @@ Calculate the Gaunt coefficient which is defined as the integral over three spherical harmonics: - \begin{eqnarray} - Y{j_1 \atop m_1} {j_2 \atop m_2} {j_3 \atop m_3} - &=& - \int Y_{l_1,m_1}(\Omega) - Y_{l_2,m_2}(\Omega) Y_{l_3,m_3}(\Omega)\; d\Omega \\ - &=& - \sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} - \left({l_1 \atop 0 } {l_2 \atop 0 } {l_3 \atop 0} \right) - \left({l_1 \atop m_1} {l_2 \atop m_2} {l_3 \atop m_3} \right) - \end{eqnarray} + Y(j_1,j_2,j_3,m_1,m_2,m_3) + =\int Y_{l_1,m_1}(\Omega) + Y_{l_2,m_2}(\Omega) Y_{l_3,m_3}(\Omega) d\Omega + =\sqrt((2l_1+1)(2l_2+1)(2l_3+1)/(4\pi)) + Y(j_1,j_2,j_3,0,0,0) Y(j_1,j_2,j_3,m_1,m_2,m_3) NOTES: @@ -773,25 +726,16 @@ The Gaunt coefficient obeys the following symmetry rules: - invariant under any permutation of the columns - \begin{eqnarray} - Y{j_1 \atop m_1} {j_2 \atop m_2} {j_3 \atop m_3} - &=& - Y{j_3 \atop m_3} {j_1 \atop m_1} {j_2 \atop m_2} - =Y{j_2 \atop m_2} {j_3 \atop m_3} {j_1 \atop m_1} - \qquad \mbox{cyclic} \\ - &=& - Y{j_3 \atop m_3} {j_2 \atop m_2} {j_1 \atop m_1} - =Y{j_1 \atop m_1} {j_3 \atop m_3} {j_2 \atop m_2} - =Y{j_2 \atop m_2} {j_1 \atop m_1} {j_3 \atop m_3} - \qquad\mbox{anti-cyclic} - \end{eqnarray} + Y(j_1,j_2,j_3,m_1,m_2,m_3) + =Y(j_3,j_1,j_2,m_3,m_1,m_2) + =Y(j_2,j_3,j_1,m_2,m_3,m_1) + =Y(j_3,j_2,j_1,m_3,m_2,m_1) + =Y(j_1,j_3,j_2,m_1,m_3,m_2) + =Y(j_2,j_1,j_3,m_2,m_1,m_3) - invariant under space inflection, i.e. - \begin{eqnarray} - Y{j_1 \atop m_1} {j_2 \atop m_2} {j_3 \atop m_3} - = - Y{j_1 \atop -m_1} {j_2 \atop -m_2} {j_3 \atop -m_3} - \end{eqnarray} + Y(j_1,j_2,j_3,m_1,m_2,m_3) + =Y(j_1,j_2,j_3,-m_1,-m_2,-m_3) - symmetric with respect to the 72 Regge symmetries as inherited for the $3j$ symbols [Regge58]